54.3.279 problem 1295

Internal problem ID [12574]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1295
Date solved : Friday, October 03, 2025 at 03:38:23 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} a \,x^{2} y^{\prime \prime }+b x y^{\prime }+\left (c \,x^{2}+d x +f \right ) y&=0 \end{align*}

✓ Maple. Time used: 0.103 (sec). Leaf size: 102

ode:=a*x^2*diff(diff(y(x),x),x)+b*x*diff(y(x),x)+(c*x^2+d*x+f)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = x^{-\frac {b}{2 a}} \left (\operatorname {WhittakerM}\left (-\frac {i d}{2 \sqrt {a}\, \sqrt {c}}, \frac {\sqrt {a^{2}+\left (-2 b -4 f \right ) a +b^{2}}}{2 a}, \frac {2 i \sqrt {c}\, x}{\sqrt {a}}\right ) c_1 +\operatorname {WhittakerW}\left (-\frac {i d}{2 \sqrt {a}\, \sqrt {c}}, \frac {\sqrt {a^{2}+\left (-2 b -4 f \right ) a +b^{2}}}{2 a}, \frac {2 i \sqrt {c}\, x}{\sqrt {a}}\right ) c_2 \right ) \]

✓ Mathematica. Time used: 0.224 (sec). Leaf size: 239

ode=(f + d*x + c*x^2)*y[x] + b*x*D[y[x],x] + a*x^2*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \left (c_1 \operatorname {HypergeometricU}\left (\frac {a+\frac {i d \sqrt {a}}{\sqrt {c}}+\sqrt {a^2-2 (b+2 f) a+b^2}}{2 a},\frac {a+\sqrt {a^2-2 (b+2 f) a+b^2}}{a},\frac {2 i \sqrt {c} x}{\sqrt {a}}\right )+c_2 L_{-\frac {a+\frac {i d \sqrt {a}}{\sqrt {c}}+\sqrt {a^2-2 (b+2 f) a+b^2}}{2 a}}^{\frac {\sqrt {a^2-2 (b+2 f) a+b^2}}{a}}\left (\frac {2 i \sqrt {c} x}{\sqrt {a}}\right )\right ) \exp \left (\int _1^x\frac {a-2 i \sqrt {c} K[1] \sqrt {a}-b+\sqrt {a^2-2 (b+2 f) a+b^2}}{2 a K[1]}dK[1]\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
d = symbols("d") 
f = symbols("f") 
y = Function("y") 
ode = Eq(a*x**2*Derivative(y(x), (x, 2)) + b*x*Derivative(y(x), x) + (c*x**2 + d*x + f)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Expected Expr or iterable but got None